Lab 04 - Turing Machines

Key concepts

1. what is a Turing-reduction?

1.1 Prove that $ R \subsetneq RE.$

1.2 Show that the following problem is not in $ R$ : $ f(M) = 1$ iff exists $ w$ such that $ M$ accepts $ w.$

• This is actually the problem complement of establishing if $ M$ accepts the non-trivial problem $ f(x) = 0.$

1.3 Show that the problem in 1.2 is in $ RE.$

1.4 Show that the following problem is in $ RE$ but not in $ R$ : $ f(M_1, M_2, w) = 1$ iff $ M_1$ and $ M_2$ both accept $ w.$

1.5 Prove that $ RE$ is countable.

2.1 Is $\leq_T$ an equivalence relation over $ RE$ ?

• $\leq_T$ is reflexive?
• $\leq_T$ is transitive?
• $\leq_T$ is symmetric? If $\leq_T$ would be symmetric, what would it signify?

2.2 Suppose $ f$ $\leq_T$ $ f_h.$ What does that say about $ f$ ?

2.3 Show that $ k$ -$ Vertex$ -$ Cover$ $\leq_T$ $ f_h.$

3.1 Show that the following problem is not in $ R$ : $ f(M_1, M_2) = 1$ iff $ M_1$ and $ M_2$ accept the same problem $ g$ (solve the same algorithm).

3.2 Show that the following problem is not in $ R$ : $ f(M) = 1$ iff $ M$ accepts all words in $ \Sigma^*$ (does $ M$ accept the problem $ g(x) = 1$ ?).

3.3 Show that the problem in 3.1 is not in $ RE.$ Use the same strategy, but now choose a problem which is not in $ RE$ for the reduction. One candidate is the $ looping$ $ problem$ from the lecture.

3.4 Show that the problem in 3.2 is not in $ RE.$

3.5 Show that the complement to the problem in 3.2 is not in $ RE.$